Klein polyhedra and relative minima of lattices (Q881027)

Let \({\mathcal L}^n\) be the space of \(n\)-dimensional unimodular lattice in \(\mathbb{R}^n\). For a lattice \(\Lambda\in{\mathcal L}^n\), by \(K_\Lambda\) denote the Klein polyhedron, and by \({\mathfrak M}(\Lambda)\) the set of all relative minima of \(\Lambda\). Moreover let \(\text{int}(K_\Lambda)\) be the set of internal points of \(K_\Lambda\), and \(\Lambda^3_{1/2}\) be the minimal lattice (with respect to inclusion) containing \(\mathbb{Z}^3\) and the point \(2^{1/3}(1/2,1/2,1/2)\). Denote by \({\mathcal L}^3_{1/2}\) the orbit of \(\Lambda^3_{1/2}\) under the left action of the group of diagonal unimodular \(3\times 3\) matrices. The author proves that if \(\Lambda\in{\mathcal L}^3\setminus{\mathcal L}^3_{1/2}\), then \({\mathfrak M}(\Lambda)\cap \text{int}(K_\Lambda)= \emptyset\). (Hence the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron.) But the similar result is false for the dimension \(n\geq 4\). Moreover, the author also proves that for almost any lattice in \(\mathbb{R}^3\), the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron.